## Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions

Optics Express, Vol. 14, Issue 24, pp. 11631-11652 (2006)

http://dx.doi.org/10.1364/OE.14.011631

Acrobat PDF (760 KB)

### Abstract

The pseudospectral method, proposed in our previous work, has not yet been constructed for optical waveguides with leaky modes or anisotropic materials. Our present study focuses on antiresonant reflecting optical waveguides (ARROWS) made by anisotropic materials. In contrast to the fields in the outermost subdomain expanded by Laguerre-Gaussian functions for guided mode problems, the fields in the high-index outermost subdomain are expanded by the Chebyshev polynomials with Mur’s absorbing boundary condition (ABC). Accordingly, the traveling waves can leak freely out of the computational window, and the desirable properties of the pseudospectral scheme, i.e., provision of fast and accurate solutions, can be preserved. A number of numerical examples tested by the present approach are shown to be in good agreement with exact data and published results achieved by other numerical methods.

© 2006 Optical Society of America

## 1. Introduction

1. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

2. T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. **6**, 1440–1445 (1988). [CrossRef]

6. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. **28**, 1689–1700 (1992). [CrossRef]

7. W. P. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. **10**, 1015–1022 (1992). [CrossRef]

8. T. Baba and Y. Kokubun, “New polarization-insensitive antiresonant reflecting optical waveguide (ARROW-B),” IEEE Photon. Technol. Lett. **1**, 232–234 (1989). [CrossRef]

7. W. P. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. **10**, 1015–1022 (1992). [CrossRef]

9. W. Jiang, J. Chrostowski, and M. Fontaine, “Analysis of ARROW waveguides,” Opt. Commun. **72**, 180–186 (1989). [CrossRef]

10. J. Chilwell and I. Hodgkinson, “Thin-film field-transfer matrix theory for planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A **1**, 742–753 (1984). [CrossRef]

13. E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. **10**, 1344–1351 (1992). [CrossRef]

14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express **7**, 260–272 (2000). [CrossRef] [PubMed]

15. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. **8**, 652–654 (1996). [CrossRef]

16. J. C. Grant, J. C. Beal, and N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. **30**, 1250–1253 (1994). [CrossRef]

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. **36**, 239–257 (2004). [CrossRef]

18. Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. **18**, 618–623 (2000). [CrossRef]

20. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. **21**, 2284–2296 (2003). [CrossRef]

20. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. **21**, 2284–2296 (2003). [CrossRef]

23. G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. **23**, 377–382 (1981). [CrossRef]

## 2. Formulations of planar waveguides

*ε̿*is expressed as follows:

*ε*

_{0}is the dielectric constant in free space, and

*n*

_{xx},

*n*

_{yy}, and

*n*

_{zz}are the refractive indices along the

*x*,

*y*, and

*z*axes of the waveguide coordinate system, respectively. In this paper, we aim at analyzing leaky modes in planar ARROW of arbitrary index profiles with a diagonal dielectric tensor shown in expression (1). Assuming that

*ε̿*is only a function of the independent

*x*(

*i.e.*, planar waveguides with finite and infinite extension in

*x*and

*y*directions, respectively), for the source-free, time-harmonic of exp(

*iωt*) form, and nonmagnetic media, the modal equations in frequency domain derived from Maxwell’s two curl equations are given as follows:

*k*

_{0}=2

*π*/

*λ*

_{0},

*λ*

_{0}denotes the wavelength in free space,

*n*

_{eff}denotes the complex effective refractive index, and

*E*

_{y}(

*x*) and

*H*

_{y}(

*x*) represent, respectively, the components of the electric and magnetic fields in the

*y*-direction.

## 3. Numerical scheme

### 3.1 Pseudospectral scheme and interfacial patching conditions

*φ*(

*x*) is expanded by a set of proper interpolation functions

*C*

_{k}(

*x*) and the unknown grid point values

*φ*

_{k}as follows:

*ψ*

_{n+1}(

*x*)denotes an arbitrary basis function of order

*n*+1 and is determined according to the feature of that subdomain, the prime denotes the first derivative of

*ψ*

_{n+1}(

*x*) with respect to

*x*, and

*x*

_{k}denotes the corresponding collocation point fulfilling the condition of

*C*

_{l}(

*x*

_{k})=

*δ*

_{lk}, where

*δ*

_{lk}denotes the Kronecker delta. The explicit forms of

*C*

_{k}(

*x*) for distinct basis functions are represented below. If we take the Chebyshev polynomials as basis functions, we have [19]

*T*

_{n}(

*x*) denotes the Chebyshev polynomial of order

*n*,

*c*

_{0}=

*c*

_{n}=2, and

*c*

_{k}=1 (1≤

*k*≤

*n*-1). For the LG functions, namely

*ψ*

_{n+1}(

*αx*)=exp(-

*αx*/2)(

*αx*)

*L*

_{n}(

*αx*) where

*L*

_{n}(

*αx*)denotes the Laguerre polynomial of order

*n*, the explicit form of

*C*

_{k}(

*x*)is given as follows [19]:

*α*called as a scaling factor that can influence the accuracy for a given number of terms of basis functions. The definite procedure for determining

*α*has been proposed in detail in our previous work [20

20. C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. **21**, 2284–2296 (2003). [CrossRef]

21. C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. **11**, 457–465 (2005). [CrossRef]

**I**denotes a unit matrix,

**Φ**denotes the vector of unknown optical field, and

**A**denotes the transverse differential operator. For obtaining the entries of the transverse operator to the

*e*th subdomain, we substitute Eq. (4) with a suitable basis function into Eqs. (2) and (3) and obtain

*x*

_{r}.

### 3.2 Determining basis functions and boundary conditions

**21**, 2284–2296 (2003). [CrossRef]

**21**, 2284–2296 (2003). [CrossRef]

23. G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. **23**, 377–382 (1981). [CrossRef]

*i. e.*, outmost collocation point), which allows the fields to radiate freely out of the computational window. The perfectly matched layer (PML) condition may be a more efficient scheme than Mur’s ABC, but the implementation of PML to the present method is very complicated. In this paper, we mainly aim to validate the accuracy and efficiency of the pseudospectral approach to leaky mode problems. As a result, we prefer choosing a simpler boundary condition like Mur’s ABC to accomplish the work.

*φ*denotes the unknown field and is expanded by the Chebyshev polynomials,

*r*denotes the outward direction normal to the boundary, and

*k*

_{r}denotes the wavenumber in

*r*-direction. In the paper,

*r*and

*k*

_{r}can be replaced by

*x*and

*k*

_{x}in the present coordinate system, respectively. Further, Eq. (13) can be expanded as follows:

*x*

_{bp}denotes the boundary point in the outermost subdomain. Note that the chosen computational interval of the outermost subdomain expanded by the Chebyshev polynomials slightly affects the convergence. Observing Eq. (14), we find that the plane wave solution of

*φ*(

*x*)=

*φ*(0)exp(-

*ik*

_{x}

*x*)is fulfilled [17

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. **36**, 239–257 (2004). [CrossRef]

*φ*for both TE and TM polarizations. As a result, Eq. (14) can be further modified as follows:

*k*

_{x}is positive. In numerical implementation, in order to keep the unit matrix

**I**in Eq. (8), the term (

*φ*is added simultaneously in two sides of Eq. (17). Therefore a matrix eigenvalue problem is obtained by evading a generalized matrix eigenvalue problem. Since the introduction of the Mur’s ABC, the eigenvalue

*n*

_{eff}resides within the global matrix and is required to be solved by iteration scheme. The computational procedure is that we first assign the initial guess of

**A**, and then a new value of

*n*

_{eff}between the adjacent iterations are simultaneously less than the order of 10

^{-6}.

17. H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. **36**, 239–257 (2004). [CrossRef]

**21**, 2284–2296 (2003). [CrossRef]

## 4. Numerical results and discussion

6. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. **28**, 1689–1700 (1992). [CrossRef]

*et al.*[14

14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express **7**, 260–272 (2000). [CrossRef] [PubMed]

### 4.1 Isotropic ARROW structure

_{a}=1for air and n

_{s}=3.85 for substrate, and the thicknesses (refractive indices) for the first cladding, second cladding, and core layers are, respectively, d

_{1}=0.142λ (n

_{1}=2.3), d

_{2}=3.15λ (n

_{2}=1.46), and d

_{c}=6.3λ (n

_{c}=1.46) under the antiresonant condition. The computational domain is divided into five subdomains. The calculated results of the complex effective indices for the first

*i*time iterations

_{a}=10 while the computational interval estimated according to our criterion [21

21. C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. **11**, 457–465 (2005). [CrossRef]

*S*

_{a}=1

*µm*, the number of Chebyshev polynomials for the first cladding, second cladding, and core layers are N

_{1}=N

_{2}=N

_{c}=20, and the Chebyshev polynomials for substrate layer is N

_{s}=40 while the computational interval of the outermost subdomain chosen is

*S*

_{sub}=1

*µm*. More terms of basis functions in the substrate layer are necessary due to the sharply oscillatory behavior of leaky waves. In addition, considering the effect caused by the computational interval of the outermost subdomain with higher refractive index, the differences of the convergent results between

*S*

_{sub}=1

*µm*and

*S*

_{sub}=2

*µm*for all of the modes are negligible while using N

_{s}=50 for

*S*

_{sub}=2

*µm*. We conclude that the major factor in obtaining accurate results is use of the pseudospectral scheme. Table 1 also shows that with only three iterations, our method achieves the convergent values. In addition, the exact four-digit values for n

_{eff}can be obtained even when only one iteration is executed. In our numerical experiments, the present approach obtains the same convergent rate for using different

_{c}. The solutions of this work, obtained by the third iteration, and the results calculated by Chen

*et al.*[14

14. C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express **7**, 260–272 (2000). [CrossRef] [PubMed]

_{a}=10, N

_{1}=N

_{2}=N

_{c}=20, and N

_{s}=40.

_{1}-TE

_{6}, versus the thickness of the first cladding (d

_{1}/λ), respectively, while other parameters are fixed. Here, referring to Ref. [6

6. T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. **28**, 1689–1700 (1992). [CrossRef]

_{0}as the first cladding mode. In Fig. 2(a), the ranges of d

_{1}/λ with flat portions of effective index for each mode correspond to the antiresonances of Fabry-Perot cavities formed in the two interference claddings. The same portions represent the low-loss portions while referring to Fig. 2(b). It is clear that the low-loss portion of TE1 is broad so as to allow large fabrication tolerance. In contrast to the flat portions, the transitional portions in Fig. 2(a) illustrate the resonances subject to high-loss portions in Fig. 2(b). For comparing the order of losses between different polarized waves, Fig. 2(b) also illustrates the first two ARROW modes of TM polarization. It is clear that the losses of TM modes are larger than TE ones at a corresponding mode number. As a result, the higher-order TE and TM modes can be easily filtered out to obtain single-mode (TE

_{1}) propagation. Besides, in Fig. 2(b), the loss characteristics display periodic variety as a function of (d

_{1}/λ). Likewise, the propagation characteristics versus the thickness of the second cladding (d

_{2}/λ) and core (d

_{c}/λ) layers are shown in Figs. 3 and 4, respectively. In Figs. 3(b) and 4(b), at corresponding first antiresonant conditions, high loss discrimination as that observed in Fig. 2(b) is also shown.

*i. e.*, d

_{1}=0.142λ, d

_{2}=3.15λ, and d

_{c}=6.3λ). Here, “relative” means that the field profiles are normalized by their maximum values [6

**28**, 1689–1700 (1992). [CrossRef]

_{0}(TM

_{0}) to TE

_{6}(TM

_{6}). We can observe that the fields in Fig. 5(a) are confined to the vicinity of the first cladding layer, and the confinement of the TE mode is tighter than that of the TM mode. Figure 5(b) illustrates that the lowest ARROW mode (TE

_{1}) is most localized in the core layer, which is of practical interest, and that it is similar to the fundamental mode of conventional waveguides. The oscillatory characteristics in the substrate layer are also clearly shown, and the larger loss of TM mode is observed, as well as the values listed in Table 2. From Figs. 5(b)–5(f), excluding Figs. 5(b) and 5(e), the fields penetrating the interference claddings are fairly large.

### 4.2 Anisotropic ARROW structure

_{zzc}=1.46 for the core, n

_{zz1}=2.3for the first cladding, and n

_{zz2}=1.46 for the second cladding. Other values follow the relations of n

_{xxj}=n

_{yyj}=1.03n

_{zzj}, where (j=c, 1,2). Here, the initial guess of effective index of

_{a}=10,N

_{1}=N

_{2}=N

_{c}=30,N

_{s}=40), are shown in Table 3 with the six-order accuracy of the finite element method [17

**36**, 239–257 (2004). [CrossRef]

**7**, 260–272 (2000). [CrossRef] [PubMed]

**36**, 239–257 (2004). [CrossRef]

**36**, 239–257 (2004). [CrossRef]

**7**, 260–272 (2000). [CrossRef] [PubMed]

### 4.3 ARROW-based directional coupler

24. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. **3**, 1135–1146 (1985). [CrossRef]

25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. **16**, 805–807 (1991). [CrossRef] [PubMed]

25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. **16**, 805–807 (1991). [CrossRef] [PubMed]

_{s}=3.5 consists of two ARROW structures separated by a separation cladding layer with a refractive index n

_{sep}=1.46 and thickness d

_{sep}=2µm [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. **14**, 1507–1513 (1996). [CrossRef]

_{c1}=n

_{c2}=1.46 and d

_{c1}=d

_{c2}=4µm. The core 1 is sandwiched by the two thin cladding layers with n

_{h1}=n

_{h2}=2.3 and d

_{h1}=d

_{h2}=0.089µm, and the same circumstance is also encountered by core 2, which is sandwiched by two cladding layers with n

_{h3}=n

_{h4}=2.3and d

_{h3}=d

_{h4}=0.089µm. For the upper cladding layer, which is grown over the thin cladding layer of d

_{h1}and in contact with air having a n

_{a}=1, the parameters are n

_{11}=1.46 and d

_{11}=2µm. As for the lower cladding layer grown over the Si substrate, it is with n

_{12}=1.46 and d

_{12}=2µm. The purpose of the different interference claddings is to accomplish the antiresonant conditions. According to the conclusion found by Chen and Huang [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. **14**, 1507–1513 (1996). [CrossRef]

_{11}. If we consider the conditions of d

_{11}=d

_{12}=2µm and d

_{11}=0,d

_{h1}=0.03µm, the maximum coupling efficiency of C

_{o}=99.87% (while the coupling length is L

_{c}=λ/(N

_{s}-N

_{a})=59mm, where N

_{s}and N

_{a}denote lowest order symmetric (even) and asymmetric (odd) modes, respectively) and decoupled phenomenon can be achieved [26

26. Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. **14**, 1507–1513 (1996). [CrossRef]

**14**, 1507–1513 (1996). [CrossRef]

_{a}=10, of Chebyshev polynomials for substrate is N

_{s}=40, and of others are N

_{i}=20 (where

*i*represents other layers). Compared with the coupling length of L

_{c}=59mm, obtained in Ref. [26

**14**, 1507–1513 (1996). [CrossRef]

_{c}=58.60mm (where N

_{s}=1.45785857 and N

_{a}=1.45785317). The coupling length of TM mode obtained by our scheme is L

_{c}=9.51mm (where N

_{s}=1.45787059 and N

_{a}=1.45783732), which is far smaller than that of the TE mode. This is because the coupling resonances of leaky waves of TM modes are larger than that of TE modes. In our calculation, the imaginary parts (radiation loss) of complex effective indices of TM modes are two orders larger than that of TE modes (namely, TE:≈10

^{-6}dB/cm, TM:≈10

^{-4}dB/cm). From the dispersion characteristics of TE modes illustrated in Fig. 7 it can be clearly seen that the maximum coupling is located at d

_{11}=2µm, making the dual ARROW acts as a symmetric coupler, and the decoupling portions are close to d

_{11}=0µm and d

_{11}=4µm. Additionally, the field profiles of the even and odd TE modes for three cases of d

_{11}=2µm (maximum coupling), d

_{11}=0.5µm (half coupling), and d

_{11}=0 (decoupling, while d

_{h1}=0.03

*µ*m is used [26

**14**, 1507–1513 (1996). [CrossRef]

25. M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. **16**, 805–807 (1991). [CrossRef] [PubMed]

**16**, 805–807 (1991). [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO |

2. | T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, “Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration,” J. Lightwave Technol. |

3. | M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on antiresonant reflecting optical waveguide,” Opt. Lett. |

4. | Z. M. Mao and W. P. Huang, “An ARROW optical wavelength filter: design and analysis,” J. Lightwave Technol. |

5. | F. Prieto, A. Llobera, D. Jimenez, C. Domenguez, A. Calle, and L. M. Lechuga, “Design and analysis of silicon antiresonant reflecting optical waveguides for evanescent field sensor,” J. Lightwave Technol. |

6. | T. Baba and Y. Kokubun, “Dispersion and radiation loss characteristics of antiresonant reflecting optical waveguides-numerical results and analytical expressions,” IEEE J. Quantum Electron. |

7. | W. P. Huang, R. M. Shubair, A. Nathan, and Y. L. Chow, “The modal characteristics of ARROW structures,” J. Lightwave Technol. |

8. | T. Baba and Y. Kokubun, “New polarization-insensitive antiresonant reflecting optical waveguide (ARROW-B),” IEEE Photon. Technol. Lett. |

9. | W. Jiang, J. Chrostowski, and M. Fontaine, “Analysis of ARROW waveguides,” Opt. Commun. |

10. | J. Chilwell and I. Hodgkinson, “Thin-film field-transfer matrix theory for planar multilayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A |

11. | J. Kubica, D. Uttamchandani, and B. Culshaw, “Modal propagation within ARROW waveguides,” Opt. Commun. |

12. | J. M. Kubica, “Numerical analysis of InP/InGaAsP ARROW waveguides using transfer matrix approach,” J. Lightwave Technol. |

13. | E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. |

14. | C. K. Chen, P. Berini, D. Feng, S. Tanev, and V. P. Tzolov, “Efficient and accurate numerical analysis of multilayer planar waveguides in lossy anisotropic media,” Opt. Express |

15. | W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer boundary conditions for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. |

16. | J. C. Grant, J. C. Beal, and N. J. P. Frenette, “Finite element analysis of the ARROW leaky optical waveguide,” IEEE J. Quantum Electron. |

17. | H. P. Uranus, H. J. W. M. Hoekstra, and E. V. Groesen, “Simple high-order Galerkin finite scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides,” Opt. Quantum Electron. |

18. | Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. |

19. | J. P. Boyd, “Chebyshev and Fourier Spectral methods,” in |

20. | C. C. Huang, C. C. Huang, and J. Y. Yang, “An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition,” J. Lightwave Technol. |

21. | C. C. Huang, C. C. Huang, and J. Y. Yang, “A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles,” IEEE J. Sel. Top. Quantum Electron. |

22. | C. C. Huang and C. C. Huang, “An efficient and accurate semivectorial spectral collocation method for analyzing polarized modes of rib waveguides,” J. Lightwave Technol. |

23. | G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. |

24. | A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. |

25. | M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, “Directional coupler based on an antiresonant reflecting optical waveguide,” Opt. Lett. |

26. | Y. H. Chen and Y. T. Huang, “Coupling-efficiency analysis and control of dual antiresonant reflecting optical waveguides,” J. Lightwave Technol. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.7390) Optical devices : Waveguides, planar

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 18, 2006

Revised Manuscript: October 23, 2006

Manuscript Accepted: October 31, 2006

Published: November 27, 2006

**Citation**

Chia-Chien Huang, "Numerical calculations of ARROW structures by pseudospectral approach with Mur’s absorbing boundary conditions," Opt. Express **14**, 11631-11652 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11631

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### References

- M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986). [CrossRef]
- T. Baba, Y. Kokubun, T. Sakaki, and K. Iga, "Loss reduction of an ARROW waveguide in shorter wavelength and its stack configuration," J. Lightwave Technol. 6, 1440-1445 (1988). [CrossRef]
- M. Mann, U. Trutschel, C. Wachter, L. Leine, and F. Lederer, "Directional coupler based on antiresonant reflecting optical waveguide," Opt. Lett. 16, 805-807 (1991). [CrossRef] [PubMed]
- Z. M. Mao and W. P. Huang, "An ARROW optical wavelength filter: design and analysis," J. Lightwave Technol. 11, 1183-1188 (1992). [CrossRef]
- F. Prieto, A. Llobera, D. Jimenez, C. Domenguez, A. Calle, and L. M. Lechuga, "Design and analysis of silicon antiresonant reflecting optical waveguides for evanescent field sensor," J. Lightwave Technol. 18, 966-972 (2000). [CrossRef]
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